576edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|576}}
{{ED intro}}


== Theory ==
== Theory ==
576 is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is what's known as a [[oeis:A033833|highly factorable edo]] and is best played through JI-agnostic approaches that make use of its divisors (see Subsets and supersets section below). This approach may be preferrable since the patent val will create sequences that fall aside by 1\576 of each other, which may not "live up to the spirit" of a composite number like 576.  
576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4|5]] is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the [[septimal ennealimma]], assigning [[7/6]] to 2\9, as well as {{monzo| 99 -66 0 2 }}, {{monzo| 110 -57 0 -7 }} , and {{monzo| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.


Nonetheless, 576edo does offer simple interpretations. Despite having bad 5/4, 576edo is [[consistent]] in the 7-odd-limit. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. Using the patent val, it tempers out the [[septimal ennealimma]], 40353607/40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from [[9edo]]. However, other commas being tempered out are far more complex – {{monzo| 99 -66 2 }}, {{monzo| 110 -57 -7 }}, and {{monzo| 88 -75 11 }}. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.
In higher limits, the 2.3.7 subgroup can be used with optional additions of [[19/16|19]] or [[29/16|29]], or fractional subgroups using [[13/10]].


In the 5-limit, 576edo supports the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val| 576 913 1336 1618 }}, is a tuning for the [[Garibaldi temperament|garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.
=== Prime harmonics ===
{{Harmonics in equal|576|columns=11}}


576edo supports a messed-up version of the [[Rectified Hebrew]] scale, but with step hardness of 5:3 instead of 3:2, and from regular temperament theory perspective, 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.
=== Subsets and supersets ===
=== Subsets and supersets ===
Its xenharmonic divisors (that is, besides 12edo and its subsets) are {{EDOs| 8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}. Some of these have been put into practical use. 72edo has been used in [[wikipedia: Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia: Alois Hába|Alois Haba]] and [[Ivan Wyschnegradsky]], and has been used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]]. Because of composition, it may be preferrable to make references to smaller edos instead of using the best approximation.
Since 576 factors as {{factorization|576}}, 576edo has subset edos {{EDOs| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a [[Highly composite equal division#Highly factorable numbers|highly factorable]] edo.
=== Prime harmonics ===
{{Harmonics in equal|576|columns=11}}


[[Category:Equal divisions of the octave|###]]
[[1152edo]], which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.


== Regular temperament properties ==
== Regular temperament properties ==
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 32: Line 30:
| 339.583
| 339.583
| 243/200
| 243/200
| [[Amity]]
| [[Amity]] (576)
|-
|-
| 12
| 12
| 239\576<br>(1\576)
| 239\576<br />(1\576)
| 497.916<br>(2.083)
| 497.916<br />(2.083)
| 4/3<br>(32805/32768)
| 4/3<br />(32805/32768)
| [[Atomic]]
| [[Atomic]] (576)
|}
|}
 
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<!-- 3-digit number -->
 
 
[[Category:Amity]]
[[Category:Atomic]]

Latest revision as of 15:45, 10 April 2026

← 575edo 576edo 577edo →
Prime factorization 26 × 32
Step size 2.08333 ¢ 
Fifth 337\576 (702.083 ¢)
Semitones (A1:m2) 55:43 (114.6 ¢ : 89.58 ¢)
Consistency limit 7
Distinct consistency limit 7

576 equal divisions of the octave (abbreviated 576edo or 576ed2), also called 576-tone equal temperament (576tet) or 576 equal temperament (576et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 576 equal parts of about 2.08 ¢ each. Each step represents a frequency ratio of 21/576, or the 576th root of 2.

Theory

576edo is consistent in the 7-odd-limit, though the error on harmonic 5 is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the septimal ennealimma, assigning 7/6 to 2\9, as well as [99 -66 0 2⟩, [110 -57 0 -7⟩ , and [88 -75 0 11⟩. In the 5-limit, the patent val of 576edo supports the atomic temperament and the amity temperament. The 576c val supports maquila. The 576ccd val, ⟨576 913 1336 1618], is a tuning for the garibaldi temperament in the 7-limit. In addition, in this case 5/4 comes from 72edo, and 7/4 comes form 288edo.

In higher limits, the 2.3.7 subgroup can be used with optional additions of 19 or 29, or fractional subgroups using 13/10.

Prime harmonics

Approximation of prime harmonics in 576edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.128 -0.897 -0.076 +0.765 -0.944 -0.789 +0.404 +0.892 -0.411 +0.798
Relative (%) +0.0 +6.2 -43.1 -3.6 +36.7 -45.3 -37.9 +19.4 +42.8 -19.7 +38.3
Steps
(reduced) 		576
(0) 		913
(337) 		1337
(185) 		1617
(465) 		1993
(265) 		2131
(403) 		2354
(50) 		2447
(143) 		2606
(302) 		2798
(494) 		2854
(550)

Subsets and supersets

Since 576 factors as 26 × 32, 576edo has subset edos 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288, of which 12, 24, 72, and 96 are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a highly factorable edo.

1152edo, which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 163\576 339.583 243/200 Amity (576)
12 239\576
(1\576) 		497.916
(2.083) 		4/3
(32805/32768) 		Atomic (576)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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